Integrand size = 24, antiderivative size = 63 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (a+b x^3\right )^2} \, dx=-\frac {c \sqrt {c+d x^3} \operatorname {AppellF1}\left (-\frac {1}{3},2,-\frac {3}{2},\frac {2}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 x \sqrt {1+\frac {d x^3}{c}}} \] Output:
-c*(d*x^3+c)^(1/2)*AppellF1(-1/3,2,-3/2,2/3,-b*x^3/a,-d*x^3/c)/a^2/x/(1+d* x^3/c)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(190\) vs. \(2(63)=126\).
Time = 10.15 (sec) , antiderivative size = 190, normalized size of antiderivative = 3.02 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (a+b x^3\right )^2} \, dx=\frac {-20 a \left (c+d x^3\right ) \left (3 a c+4 b c x^3-a d x^3\right )+5 c (-8 b c+11 a d) x^3 \left (a+b x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+2 d (4 b c-a d) x^6 \left (a+b x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )}{60 a^3 x \left (a+b x^3\right ) \sqrt {c+d x^3}} \] Input:
Integrate[(c + d*x^3)^(3/2)/(x^2*(a + b*x^3)^2),x]
Output:
(-20*a*(c + d*x^3)*(3*a*c + 4*b*c*x^3 - a*d*x^3) + 5*c*(-8*b*c + 11*a*d)*x ^3*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c) , -((b*x^3)/a)] + 2*d*(4*b*c - a*d)*x^6*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*Ap pellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)])/(60*a^3*x*(a + b*x^3 )*Sqrt[c + d*x^3])
Time = 0.36 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1013, 1012}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (a+b x^3\right )^2} \, dx\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {c \sqrt {c+d x^3} \int \frac {\left (\frac {d x^3}{c}+1\right )^{3/2}}{x^2 \left (b x^3+a\right )^2}dx}{\sqrt {\frac {d x^3}{c}+1}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle -\frac {c \sqrt {c+d x^3} \operatorname {AppellF1}\left (-\frac {1}{3},2,-\frac {3}{2},\frac {2}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 x \sqrt {\frac {d x^3}{c}+1}}\) |
Input:
Int[(c + d*x^3)^(3/2)/(x^2*(a + b*x^3)^2),x]
Output:
-((c*Sqrt[c + d*x^3]*AppellF1[-1/3, 2, -3/2, 2/3, -((b*x^3)/a), -((d*x^3)/ c)])/(a^2*x*Sqrt[1 + (d*x^3)/c]))
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ n/a))^FracPart[p]) Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 0])
Result contains higher order function than in optimal. Order 9 vs. order 6.
Time = 4.81 (sec) , antiderivative size = 970, normalized size of antiderivative = 15.40
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(970\) |
risch | \(\text {Expression too large to display}\) | \(1854\) |
default | \(\text {Expression too large to display}\) | \(2364\) |
Input:
int((d*x^3+c)^(3/2)/x^2/(b*x^3+a)^2,x,method=_RETURNVERBOSE)
Output:
1/3*(a*d-b*c)/a^2*x^2*(d*x^3+c)^(1/2)/(b*x^3+a)-c/a^2*(d*x^3+c)^(1/2)/x-2/ 3*I*(-1/6*d*(a*d-b*c)/a^2/b+1/2*d*c/a^2)*3^(1/2)/d*(-c*d^2)^(1/3)*(I*(x+1/ 2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3 ))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(- c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^ (1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*((-3/2/d*(-c*d^2)^( 1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/d*(-c *d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2 ),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d ^2)^(1/3)))^(1/2))+1/d*(-c*d^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(- c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/ 2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c* d^2)^(1/3)))^(1/2)))+1/18*I/a^2/b/d^2*2^(1/2)*sum((-a^2*d^2-7*a*b*c*d+8*b^ 2*c^2)/_alpha/(a*d-b*c)*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d ^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3)) /(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I *3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^( 1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/2)*(-c*d^2)^(2/3)+2*_alpha^ 2*d^2-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x +1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2...
Timed out. \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (a+b x^3\right )^2} \, dx=\text {Timed out} \] Input:
integrate((d*x^3+c)^(3/2)/x^2/(b*x^3+a)^2,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (a+b x^3\right )^2} \, dx=\text {Timed out} \] Input:
integrate((d*x**3+c)**(3/2)/x**2/(b*x**3+a)**2,x)
Output:
Timed out
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (a+b x^3\right )^2} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (b x^{3} + a\right )}^{2} x^{2}} \,d x } \] Input:
integrate((d*x^3+c)^(3/2)/x^2/(b*x^3+a)^2,x, algorithm="maxima")
Output:
integrate((d*x^3 + c)^(3/2)/((b*x^3 + a)^2*x^2), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (a+b x^3\right )^2} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (b x^{3} + a\right )}^{2} x^{2}} \,d x } \] Input:
integrate((d*x^3+c)^(3/2)/x^2/(b*x^3+a)^2,x, algorithm="giac")
Output:
integrate((d*x^3 + c)^(3/2)/((b*x^3 + a)^2*x^2), x)
Timed out. \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (a+b x^3\right )^2} \, dx=\int \frac {{\left (d\,x^3+c\right )}^{3/2}}{x^2\,{\left (b\,x^3+a\right )}^2} \,d x \] Input:
int((c + d*x^3)^(3/2)/(x^2*(a + b*x^3)^2),x)
Output:
int((c + d*x^3)^(3/2)/(x^2*(a + b*x^3)^2), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (a+b x^3\right )^2} \, dx =\text {Too large to display} \] Input:
int((d*x^3+c)^(3/2)/x^2/(b*x^3+a)^2,x)
Output:
( - 2*sqrt(c + d*x**3)*c + 2*int((sqrt(c + d*x**3)*x**4)/(a**3*c*d + a**3* d**2*x**3 - 8*a**2*b*c**2 - 6*a**2*b*c*d*x**3 + 2*a**2*b*d**2*x**6 - 16*a* b**2*c**2*x**3 - 15*a*b**2*c*d*x**6 + a*b**2*d**2*x**9 - 8*b**3*c**2*x**6 - 8*b**3*c*d*x**9),x)*a**3*d**3*x - 21*int((sqrt(c + d*x**3)*x**4)/(a**3*c *d + a**3*d**2*x**3 - 8*a**2*b*c**2 - 6*a**2*b*c*d*x**3 + 2*a**2*b*d**2*x* *6 - 16*a*b**2*c**2*x**3 - 15*a*b**2*c*d*x**6 + a*b**2*d**2*x**9 - 8*b**3* c**2*x**6 - 8*b**3*c*d*x**9),x)*a**2*b*c*d**2*x + 2*int((sqrt(c + d*x**3)* x**4)/(a**3*c*d + a**3*d**2*x**3 - 8*a**2*b*c**2 - 6*a**2*b*c*d*x**3 + 2*a **2*b*d**2*x**6 - 16*a*b**2*c**2*x**3 - 15*a*b**2*c*d*x**6 + a*b**2*d**2*x **9 - 8*b**3*c**2*x**6 - 8*b**3*c*d*x**9),x)*a**2*b*d**3*x**4 + 40*int((sq rt(c + d*x**3)*x**4)/(a**3*c*d + a**3*d**2*x**3 - 8*a**2*b*c**2 - 6*a**2*b *c*d*x**3 + 2*a**2*b*d**2*x**6 - 16*a*b**2*c**2*x**3 - 15*a*b**2*c*d*x**6 + a*b**2*d**2*x**9 - 8*b**3*c**2*x**6 - 8*b**3*c*d*x**9),x)*a*b**2*c**2*d* x - 21*int((sqrt(c + d*x**3)*x**4)/(a**3*c*d + a**3*d**2*x**3 - 8*a**2*b*c **2 - 6*a**2*b*c*d*x**3 + 2*a**2*b*d**2*x**6 - 16*a*b**2*c**2*x**3 - 15*a* b**2*c*d*x**6 + a*b**2*d**2*x**9 - 8*b**3*c**2*x**6 - 8*b**3*c*d*x**9),x)* a*b**2*c*d**2*x**4 + 40*int((sqrt(c + d*x**3)*x**4)/(a**3*c*d + a**3*d**2* x**3 - 8*a**2*b*c**2 - 6*a**2*b*c*d*x**3 + 2*a**2*b*d**2*x**6 - 16*a*b**2* c**2*x**3 - 15*a*b**2*c*d*x**6 + a*b**2*d**2*x**9 - 8*b**3*c**2*x**6 - 8*b **3*c*d*x**9),x)*b**3*c**2*d*x**4 + 5*int((sqrt(c + d*x**3)*x)/(a**3*c*...